In terms of when's the first opportunity for teams to have a crack, there's too many permutations to figure them all out.
However, from what I can tell, Brisbane City will have first crack at the Shield regardless of who wins, either Round 2 against the Rising, or at Country's next home game in Round 3, assuming they wouldn't accept the Stars challenge.
If the Rising win the first 2 games, all their remaining fixtures will be challenges, and if they continued to win, that would be the first date available to get the hands on the shield.
As far as I can tell, the quickest possible way for the Rams to get their hands on the Shield, (to answer your question Pfitzy) would be for Queensland Country to win it off Rising in round 1, accept and win Challenges in round 2 and 3 against the Stars and Brisbane City, then play the Rams for it in round 4.
The next best option would be for the Rising to win all their challenges, and you play off against them in round 5, which is a home rising fixtures and wouldn't rely on accepting away challanges.